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Physical Review A

Path-unbalanced fourth-order interference with the aid of a weak coherent reference

Xuan Tang, Z. Y. Ou

Published Jul 17, 2025• 0 citations
Publisher

Abstract

Traditionally, observation of optical interference in intensity requires the balance of the paths of the two interfering fields within the coherence length of the optical fields. With intensity correlation fourth-order measurement between two detectors, such a requirement can be relaxed. Here, we report the observation of interference in a path-unbalanced Mach-Zehnder interferometer by mixing the output field with a weak coherence reference field before making a fourth-order intensity correlation measurement. Detailed theoretical analysis is made and is consistent with the experimental observation. As a side product of the study, when the coherent reference field is not present, the path-unbalanced interferometer realizes a Hong-Ou-Mandel interference effect that can be used to measure the coherence time of the input field with ${g}^{(2)}=1$. Such a technique of recovering interference, which is otherwise absent with the traditional observation method, should find wide applications in optical sensing by extending the applicable range.

DOI
10.1103/46jb-lzk2

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•Jan 1, 2018•2 citations
Traditionally, observation of optical interference in intensity requires the balance of the paths of the two interfering fields within the coherence length of the optical fields. With intensity correlation fourth-order measurement between two detectors, such a requirement can be relaxed. Here, we report the observation of interference in a path-unbalanced Mach-Zehnder interferometer by mixing the output field with a weak coherence reference field before making a fourth-order intensity correlation measurement. Detailed theoretical analysis is made and is consistent with the experimental observation. As a side product of the study, when the coherent reference field is not present, the path-unbalanced interferometer realizes a Hong-Ou-Mandel interference effect that can be used to measure the coherence time of the input field with g(2)=1{g}^{(2)}=1g(2)=1. Such a technique of recovering interference, which is otherwise absent with the traditional observation method, should find wide applications in optical sensing by extending the applicable range.
Interferometry has been used widely in sensing application. However, the technique is limited by the finite coherence time of the light sources when the interference paths are not balanced. Higher-order interference effects involve intensity correlations between multiple detectors and may have the advantage over the traditional second-order interference effect exhibited in only one detector. We discuss various scenarios with unbalanced delays in different paths in fourth-order interference exhibited in coincidence between two detectors. We find, in some cases, interference effect persists even when the delays are much larger than the coherence time of the sources. We also extend the discussion to nonstationary pulsed fields, which need to consider the pulse shape and require a different treatment. These results will be useful in remote sensing applications.
Interference effects are usually observed by intensity measurement. Path indistinguishability by the quantum complementarity principle requires projection of the interfering fields into a common indistinguishable mode before detection. On the other hand, the essence of wave interference is the addition of amplitudes of the interfering fields. Therefore, if amplitudes can be directly measured and added, interference can occur even though the interfering fields are in well-distinguishable modes. Here, we make a comprehensive study in both theory and experiment of a technique by homodyne measurement of field amplitudes to reveal interference. This works for both classical and quantum fields even though there exists distinguishability in the interfering paths of light. This directly challenges the complementarity principle. We present a resolution of this issue from the viewpoint of measurement that emphasizes either particle or wave. This technique is particularly useful for recovering interference in unbalanced interferometers with path imbalance beyond a coherence length of the input field and can be applied to remote sensing to extend the applicable range. Since the amplitude-based interference phenomena studied here are fundamentally different from the traditional intensity-based interference phenomena, our approach leads to a new paradigm to study coherence between optical fields.
Optical interference usually requires two interfering paths to be balanced within the coherence length. For the case of unbalanced paths, interference effects can be recovered in the frequency-domain analysis of the superposed resultant optical field because optical frequency filtering in the spectral analysis of the optical field lengthens the coherence length to overcome the path imbalance. However, since optical detection usually is intensity measurement that loses the phase information of the detected optical field, the output photocurrent of intensity measurement cannot recover the lost interference effect in the postprocessing of the data such as electronic spectral analysis. On the other hand, if amplitude measurement can be used to measure the resultant optical field from the path-unbalanced superposition of two fields, the output photocurrent carries complete information of two fields and can restore interference in postdetection processing, even though no interference shows up in direct intensity measurement. In this paper, we use homodyne detection method for amplitude measurement to analyze the output of a path-unbalanced interferometer in the spectral domain. Since complete information about the superposed optical field is contained in the photocurrent, spectral analysis shows interference fringes in frequency that reveal information about the path imbalance. It is found that shot noise is the limiting factor in the visibility of interference but can be reduced with a squeezed state input in the unused port of the unbalanced interferometer. This approach is a different paradigm from the traditional one for studying the coherence between optical fields and should find wide application in remote sensing and precision measurement.
We report observation of a counter-intuitive phenomenon in multi-path correlation interferometry with thermal light. The intensity correlation between the outputs of two unbalanced Mach-Zehnder interferometers (UMZI) with two classically correlated beams of thermal light at the input exhibits genuine second-order interference with the visibility of 1/3. Surprisingly, the second-order interference does not degrade at all no matter how much the path length difference in each UMZI is increased beyond the coherence length of the thermal light. Moreover, the second-order interference is dependent on the difference of the UMZI phases. These results differ substantially from those of the entangled-photon Franson interferometer which exhibits two-photon interference dependent on the sum of the UMZI phases and the interference vanishes as the path length difference in each UMZI exceeds the coherence length of the pump laser. Our work offers deeper insight into the interplay between interference and coherence in multi-photon interferometry.
By using an intensity correlation, or photon coincidence, technique and a Mach-Zehnder interferometer with artificially applied phase fluctuations in one arm, we measured the coherence time of a thermal light beam. The light was derived from the filtered fluorescence of a rhodamine 6G dye solution. The two-photon coincidence rate was measured as a function of path difference and was found to dip 50% when the two arms of the interferometer are balanced. The coherence time of ~100 fs is determined from the measured width of the dip and is consistent with the passband of the filters. The method does not require the stability needed in second-order interference measurements, and its time resolution can be much shorter than that of the photodetectors. The theory of this type of fourth-order interference technique is discussed.
We report an experimental demonstration for a counter-intuitive phenomenon in multi-path interferometer with thermal light. The intensity correlation between the outputs of two unbalanced Mach-Zehnder interferometers (UMZIs) with two correlated beams of thermal light at the input exhibits genuine second-order interference with the visibility of 1/3. Here, the second-order interference does not degrade at all no matter how much the path length difference in each UMZI is increased beyond the coherence length of the thermal light.